Inferring Ecological Relationships from the Edges of Scatter Diagrams: Comparison of Regression Techniques

Scatter diagrams have historically proved useful in the study of associative relationships in ecology. Several important ecological questions involve correlations between variables resulting in polygonal shapes. Two examples that have received considerable attention are patterns between prey size and predator size in animal populations and the relationship between animal abundance and body size. Each is typically illustrated using scatter diagrams with upper and lower boundaries of response variables often changing at different rates with changes in the independent variables. Despite recent statistical contributions that have stimulated an interest in characterizing the limits of a variable, a consensus on an appropriate methodology to quantify the boundaries of scatter diagrams has not yet been achieved. We tested regression techniques based on least squares and least absolute values models using several independent data sets on prey length and predator length for piscivorous fishes and compared estimated slopes for consistency. Our results indicated that least squares regression techniques were particularly sensitive to outlying y values and irregularities in the distribution of observations, and that they frequently produced inconsistent estimates of slope for upper and lower bounds. In contrast, quantile regression techniques based on least absolute values models appeared robust to outlying y values and sparseness within data sets, while providing consistent estimates of upper and lower bound slopes. Moreover, the use of quantile regression eliminated the need for an excess of arbitrary decision-making on the part of the investigator. We recommend quantile regression as an improvement to currently available techniques used to examine potential ecological relationships dependent upon quantitative information on the boundaries of polygonal relationships.